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"J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 DOI 10.1007/s40030-013-0042-y OR IGINAL CONTRIBUTION Equivalent Strut Method for the Modelling of Masonry Inﬁll Walls in the Nonlinear Static Analysis of Buildings A. Adukadukam A. K. Sengupta Received: 1 June 2012 / Accepted: 10 October 2013 / Published online: 30 October 2013 The Institution of Engineers (India) 2013 Abstract In the seismic analysis of a framed building Keywords Equivalent strut  Inﬁll wall with masonry inﬁll walls, it is necessary to model the effect Masonry wall  Nonlinear static analysis of the walls on the lateral stiffness, strength and ductility of Pushover analysis  Seismic analysis the building. The equivalent strut method is convenient for modelling the walls in a large building. However, an List of symbols appropriate axial load versus deformation relationship for B Breadth of section the strut is required in a nonlinear static method of seismic D Depth of section analysis, such as the pushover analysis. The present study d Length of the equivalent strut, equal to the proposes a nonlinear axial hinge property for the strut, with diagonal length of the inﬁlled panel suitable performance levels. First, the equivalent strut f Allowable compressive strength along the diagonal of a wall based on slenderness ratio method and the suitability of two approaches available in the literature for modelling the properties of the struts, are f Design bond shear strength between masonry units bs brieﬂy discussed. Next, the nonlinear axial load versus and mortar deformation relationship is developed based on experi- h Height of centre-line of beam from top of footing mental data compiled from the literature. The parabolic– h Clear height of the wall plastic relationship is idealized as a tri-linear axial hinge l Centre-line to centre-line width of bay property, so that it can be incorporated in commercial l Clear length of the wall software for undertaking pushover analysis. Next, the use R Axial load in the equivalent strut of the hinge property is demonstrated in the pushover R Load in the equivalent strut at local crushing of the analyses of two framed reinforced concrete buildings. The corners of the inﬁll wall pushover curves based on the proposed hinge property R Load in the equivalent strut at shear cracking of the shows improved modelling of the inelastic drifts of the inﬁll wall buildings. Although the modelling of a wall using a single R Load in the equivalent strut at diagonal dc strut has limitations, the proposed methodology is practical compression failure of the inﬁll wall for a pushover analysis of a building. R Ultimate strength of the equivalent strut t Thickness of wall a Coefﬁcient measuring the length of the wall in contact with the column d Axial deformation of the equivalent strut A. Adukadukam e Axial strain in the equivalent strut School of Mechanical and Building Sciences, Vellore Institute of Technology, Vellore, India e Strain in the equivalent strut when the strength is attained (R/R attains the A. K. Sengupta (&) value of 1) = 0.0025 Department of Civil Engineering, Indian Institute of Technology e Strain in the equivalent strut till the strength is Madras, Chennai 600036, India retained = 0.004 e-mail: amlan@iitm.ac.in 123 100 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 member-level non-linear behaviours of the frame members e , e Strain in the equivalent strut at the diagonal 1 2 and the struts can be assembled to get the global non-linear compression failure of a slender inﬁll wall response of a building under lateral forces, using the static / Diameter of bars pushover analysis or dynamic time history analysis. The l Coefﬁcient of internal friction between masonry results can further be used in a performance based analysis units and mortar of a building, which quantiﬁes damage in terms of per- h Angle of inclination of diagonal of the panel, with formance levels, under a certain level of seismic hazard. respect to horizontal Since the present study investigated the modelling of the h Angle of inclination of strut considering the length eff inﬁll walls as equivalent struts in the pushover analysis of a of wall in contact with the column building, the equivalent strut method is brieﬂy described. r Average normal stress on wall in contact with column s Average shear stress on wall in contact with beam Equivalent Strut Method Introduction Under the action of in-plane lateral load, an inﬁlled frame behaves like a solid cantilever with an initial bond at the In a framed multi-storeyed building, the exterior enclosing frame–wall interface [11]. An inﬁll wall which is not walls and interior partition walls are considered to be inﬁll physically connected to the bounding members by shear walls if they are placed within the panels of the frames. keys or dowels, is classiﬁed as non-intergral. For such a Traditionally, in a structural analysis of the building, the wall, after the bond is broken, there is separation between inﬁll walls are treated as a non-structural architectural the frame and the wall, at the diagonally opposite corners component of the building. They are not explicitly mod- of the panel that are under tension. Compressive load is elled in a computational analysis. For heavy masonry transferred to the wall near the other corners which are walls, their weight is assigned on the supporting beams. under compression, through certain lengths of contact with However, if there is no gap or negligible gap between the the bounding frame members. The wall then behaves like a walls and the bounding frame members, the walls tend to strut under compression. have a substantial inﬂuence on the structural behaviour of Holmes [6] proposed replacing the wall by an equivalent the building during an earthquake. The modelling of inﬁll pin-jointed strut along the diagonal under compression, walls becomes imperative in the seismic analysis of a made up of the same material as that of the wall. The framed building, for the following reasons. First, the walls thickness of the strut is same as that of the wall, but the increase the lateral stiffness of the building, leading to width is equal to one-third of the length of the diagonal of larger lateral loads. Second, irregular placement of the the inﬁll. This is a minimalistic model, suitable for the walls can generate plan or vertical irregularity. The shift in analysis of buildings with several inﬁll walls in the storeys. the centre of stiffness for a ﬂoor or presence of open A theoretical relationship between the width of the strut ground storey, and the possibility of brittle shear failure of and a relative stiffness parameter of the frame—wall sys- columns next to a partial height wall remain undetected if tem has been provided by the researchers [15]. The width the effect of the walls is not modelled. Third, the of the strut was given in the form of charts for different strengthening of an open ground storey by providing inﬁll loads on the strut, and the aspect ratio of the inﬁlled panel. walls cannot be justiﬁed by analysis unless the effect of the An alternate equivalent strut model based on a plastic walls is modelled. analysis was developed earlier [8]. An equivalent strut Accurate numerical modelling of the nonlinear behav- method was proposed based on the ultimate load (UL) that iour of masonry inﬁlled frames under in-plane lateral loads, can be carried by the wall [12]. The aspect ratio of the by the ﬁnite element method is difﬁcult. The modelling inﬁlled panel, interface stresses due to the frame–wall involves consideration of the individual masonry units, interaction, and the ﬂexural capacities of the bounding mortar in the bed joints and the frame–wall interface, and beams and columns are accounted for in this method. their appropriate constitutive models [10]. Developing such The failure of an inﬁll wall in a frame under in-plane a computational model is intensive and time consuming for lateral load can be categorized into the following three the analysis of multi-storeyed buildings with multiple walls modes. The corresponding compressive loads in the in each storey. Hence, researchers have investigated alter- equivalent strut are denoted in the brackets. nate macro-scale modelling of the walls, based on the • Local crushing (R ) of the inﬁll at one of the loaded concept of equivalent struts which simulates the composite corners. behaviour of an inﬁlled frame. The modelling of the 123 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 101 • Shear cracking (R ) along the bed joints of the masonry analysis of a framed building with inﬁll walls by the units. equivalent static method (static analysis) or the response • Diagonal compression failure (R ) due to instability of spectrum method (dynamic analysis), modelling of the dc a slender inﬁll wall. walls by the simpler EA approach is adequate. In these analyses, the member forces are of primary interest. But in The diagonal tensile cracking need not be considered as a a pushover analysis (nonlinear static analysis) of a build- failure mode, as higher load can be applied beyond tensile ing, the UL approach is preferred. Here, the ultimate lateral cracking. strength and the inelastic drift of the building under The expressions for the three failure loads are provided increasing lateral load are of primary interest. Since the UL as given below [12]: approach gives a lower strut width and hence lower stiff- ness for a strut, it provides a more realistic estimate of the ð1  a Þa thr þ a tls c c c b b R ¼ ð1Þ inelastic drift. However in a pushover analysis, in addition cos h to the strut width and strength, the axial load versus f tl bs deformation behaviour of a strut is also required. R ¼ ð2Þ ð1  l tan h Þ cos h eff The modelling of an inﬁll wall by a single strut has limitations. First, it is not suitable when there is a central 0:5htf R ¼ ð3Þ dc opening in the wall. An alternate is to use multi-strut cos h model. Of course for a large opening as is common with The symbols used in the expressions are explained windows, the strut effect of the wall is negligible, due to under List of symbols. The lowest of R , R and R is the c s dc the drastic reduction in diagonal stiffness. Hence, the strut strength (R ) of the equivalent strut. The failure of a wall need not be modelled. Second, the procedure cannot primarily due to out-of-plane bending is not an effect of explicitly model the yielding of the columns or beams at the in-plane lateral load. The width of the strut is the portions in contact with the wall [3]. It has been calculated from R based on a design compressive observed that a model consisting of three struts parallel to strength for the masonry. A secant modulus of elasticity the diagonal of the wall under compression, is effective in is considered for the strut. capturing the ﬂexural behaviour of the bounding frame members. However, the three struts model needs additional effort in developing the computational model for a large Investigation of Equivalent Strut Methods building. Hence, in the present study a single strut model was adopted, that is expected to be appealing to the prac- Essentially there are two approaches in modelling the ticing professionals to get the overall behaviour of a properties of the equivalent strut for an inﬁll wall. The building under a pushover analysis. approach as proposed is the basis for a conventional linear analysis of a frame under lateral load, and has been adopted in this work [5, 15]. In this paper, this approach is referred Modelling of Axial Load against Deformation to as the elastic analysis (EA) approach. The width of a Behaviour strut is calculated based on the overall lateral stiffness of the inﬁlled frame. On the other hand, the approach as The axial load versus deformation behaviour of an equiv- proposed [12] is based on the governing failure load. It is alent strut can be considered to be linear for convenience applicable beyond the linear elastic range of the wall. In this paper, this approach is referred to as the UL approach. The procedure for calculating the width of a strut implicitly considers the yielding of the bounding columns or beams B, C near the joints. A comparative study of both the approaches in deter- mining the properties of the equivalent strut, and the applicability of the two approaches in linear and nonlinear analyses of a framed building, were reported by the D  E 0.2R researchers [14]. The EA approach is simpler in calcula- tion. A higher strut width gives higher stiffness for a strut Axial deformation and hence, higher lateral load (base shear) for a building. Since the EA approach gives higher strut width, it is con- Fig. 1 Linear axial load against deformation curve for an equivalent servative in estimating the base shear. In a linear seismic strut Axial load  102 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 Fig. 2 Typical specimen and the corresponding model Jack (a) Test specimen (b) Computational model Table 1 Properties of the frame members in the test specimens Reference l,m h, m Section dimensions (B 9 D), mm 9 mm Reinforcement details (/), mm Concrete cube strength, Beam Column Beam Column MPa Top Bottom Achintya et al. [1] 1.45 1.15 135 9 250 135 9 250 2–20 2–20 4–20 15.00 Achyutha et al. [2] 1.38 0.75 100 9 125 100 9 125 2–8 2–8 6–8 58.60 Mehrabi et al. [9] 2.31 1.52 160 9 230 180 9 180 2–16 2–16 8–12 26.80 Table 2 Properties of inﬁll / / Reference t,mm l ,m h , m Compressive strength Bond shear walls in the test specimens of masonry, MPa strength ðf Þ, MPa bs Achintya et al. [1] 115 1.20 1.00 7.50 0.24 Achyutha et al. [2] 60 1.25 0.63 14.00 1.20 Mehrabi et al. [9] 100 2.13 1.42 15.00 0.48 (Fig. 1), under the presumption that the masonry units and lateral load versus drift tests of single-bay single-storey mortar are quasi-brittle in nature, and the inﬁll walls are inﬁlled reinforced concrete (RC) frame specimens [1, 2, 9]. secondary members in a frame. However, the behaviour of For calculating the values of R, the test for each specimen an inﬁll wall under diagonal compression is nonlinear due was simulated by a computational model using the program to the formation and development of multiple types of SAP 2000 NL [13] (Fig. 2). Each of the beam and columns cracks. In the present study, emphasis was placed in was modelled by a frame element, with non-linear ﬂexural modelling the non-linear behaviour of an equivalent strut. hinge property near the joints, calculated based on the The nonlinear behaviour can be incorporated in the com- section dimensions and the reinforcement (Table 1). The putational model by assigning an axial ‘plastic hinge’ at the columns were considered to be ﬁxed at the foundations. For middle of the strut, with appropriate axial load versus a strut, the UL approach was used to calculate its width deformation curve for the hinge. The curve is referred to as from the properties given in Table 2. The thickness of the the hinge property. Although the approach is based on strut was kept same as that of the wall. At each step for the lumped plasticity, the non-linear deformation is calculated applied load H, the value of R in the strut was noted from over the length of the strut. In a performance based ana- the analysis of the model. The corresponding value of d lysis, the performance levels such as immediate occupancy was calculated as the diagonal component of the recorded (IO), life safety (LS) and collapse prevention (CP) need to drift. be deﬁned for the hinge property, to monitor the state of To get a common trend in the R versus d variations in a damage in the strut. single plot for all the analysed specimens, the individual In the present study, an axial load (R) versus deforma- values of R and d were normalised with respect to the cor- tion (d) relationship was developed for the equivalent strut, responding calculated maximum load (R ) and diagonal based on the analyses of published results of monotonic length (d), respectively. The normalised axial load (R/R ) 123 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 103 Original curve 1.2 Linear approximation 1.0 Assumed curve beyond CP 0.8 CP B D LS IO 0.6 Achintya et al., 1991 Achyutha et al., 1994 0.4 Mehrabi et al., 1996 0.2R 0.2 Proposed curve 0.5 0.0 0 0 u (0.004) (0.00125) (0.0025) 0.000 0.001 0.002 0.003 0.004 0.005 Fig. 4 Linear approximation of axial load against deformation curve Fig. 3 Nonlinear axial load against deformation curve for an equivalent strut tangential slope of the parabolic curve reduces to half the initial value. The LS performance level was selected at versus deformation (d/d) data points are shown in Fig. 3. The e = e = 0.0025, the strain corresponding to the attaining of maximum load in the strut. The CP performance level normalised deformation (d/d) is denoted as axial strain (e). Using the above data points, a curve similar to that for was selected at e = e = 0.004, beyond which the load concrete under ﬂexural compression was ﬁtted. The curve carrying capacity of the strut is uncertain and is expected to is parabolic till R/R attains unity at the strain e = 0.0026. drop. In the present study it was not attempted to trace the u 0 For convenience, in the proposed equation the strain was post-ultimate descending branch of the axial load versus taken as 0.0025, for normal strength masonry units. After deformation behaviour of the equivalent strut. A drop in this R/R is equated to unity till e = 0.004. strength was assumed at e = 0.004, representing the failure u u The proposed equations for the axial load versus of the inﬁll wall. A residual strength of 20 % of the peak deformation curve (Fig. 3) are given as follows. For e \\ e , strength was assumed based on conventional practice, to \"# avoid any numerical instability in the computation for e e pushover analysis due to zero post-peak stiffness value for R ¼ R 2  ð4aÞ e e 0 0 a member. The speciﬁc equations of the piece-wise linear curve can be derived from the coordinates of the kinks in For e \\ e B e , 0 u the curve, as follows. R ¼ R ð4bÞ For e B 0.5e , The above equations are valid when the strength is 3 e governed by shear cracking or corner crushing failure modes. R ¼ R ð5aÞ 2 e In such a case R is the lower of R and R . These are the u c s modes of material failure. When the failure is by diagonal compression, a sudden drop in the curve is expected to occur The slope of the curve represents the secant-stiffness of the due to instability of the slender inﬁll wall. This is a rare original curve up to the IO performance level. failure mode which is limited to only slender walls. For 0.5e \\ e B e , 0 0 R e R ¼ 1 þ ð5bÞ 2 e Simpliﬁed Relationships with Performance Levels For a pushover analysis of a building, instead of using the The slope of the curve represents the secant-stiffness of the nonlinear axial load versus deformation relationship, a original curve between the IO and LS performance levels. simpliﬁed piece-wise linear relationship is proposed for the For e \\ e B e , 0 u hinge property of the equivalent strut. This simpliﬁcation is R ¼ R ð5cÞ necessary for the input of a hinge property in a computa- tional model. For e [ e , Figure 4 shows the tri-linear (up to the ultimate state) R ¼ 0:2R ð5dÞ axial load versus deformation curve. The three perfor- mance levels were decided in conjunction with the FEMA 356 recommendations. The IO performance level was Instead of a discrete jump in strength at the ultimate, a gradual reduction can also be adopted. The above hinge selected at e = 0.5 e = 0.00125, the strain at which the R/Ru 104 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 compression failure. All the performance levels IO, LS and CP are deﬁned at the same point corresponding to C D the failure load R . The following equations represent B dc the hinge property for the equivalent strut. Diagonal compression failure For e B e , IO, LS, CP dc R ¼ R ð6aÞ dc 0.2R 1 dc A 0 u 1 0.5 0 0 For e [ e , (a) R within AB 1 dc R ¼ 0:2R ð6bÞ dc dc 2 R dc Here, e ¼ e . B 1 0 LS, CP 3 R IO Case 2: 0.75R \\ R \\ R u dc u Diagonal compression failure If R is between 0.75R and R , there will be a drop in dc u u the load carrying capacity in the second linear region 0.2R dc (BC). The dotted line in Fig. 5b shows the drop due to 0 u 0.5 2 0 0 diagonal compression failure. The IO performance level is deﬁned at 0.5e as before. The LS and CP performance (b) R within BC 0 dc levels are deﬁned at the same point corresponding to the Fig. 5 Axial hinge property for equivalent strut under diagonal failure load R . The following equations represent the dc compression failure, a R within AB, b R within BC dc dc hinge property for the equivalent strut. For e B 0.5e , property is applicable when the failure is governed by 3 e R ¼ R ð7aÞ corner crushing or shear cracking. In the case of diagonal 2 e compression failure, since R is lower than R (lower of R dc u c and R ), a drop in the curve is modelled when the axial load reaches R , before attaining the strength governed by R or For 0.5e \\ e B e , dc c 0 2 R . R  R u dc R ¼ R  ðÞ e  e ð7bÞ dc 2 There can be two cases based on the value of R e  e 0 2 dc compared to the load at IO performance level, which is 0.75 R For e B e , Case 1: R B 0.75R dc u R ¼ 0:2R ð7cÞ dc If R is less than 0.75R , there will be a drop in the load dc u carrying capacity in the ﬁrst linear region (AB). The dc Here, e ¼ 2  1 e . dotted line in Fig. 5a shows the drop due to diagonal 2 0 Fig. 6 Typical plan and 6m 6m 6m 6m elevation of Building I 3.6m 6m 3.6m 6m 3.6m 3.6m 6m (b) Frame elevation (a) Floor plan 123 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 105 Based on linear hinge property Based on proposed hinge property 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Roof displacement m Fig. 8 Pushover curve along X-direction (Building I) (a) Using linear hinge property IO LS CP the constitutive relationships of the two materials. A bi- linear shear hinge property was idealised based on the initial shear stiffness and ultimate shear strength. For a strut element, two alternatives of the axial hinge property were used. First, a conventional linear hinge property was used, as per the EA approach (Fig. 1). Here, all the three performance levels were considered at the peak of the curve. Next, the proposed (Eqs. 5–7) tri-linear hinge property was used. The pushover curves using the two types of hinge properties were compared to observe the effect of modelling the hinge property of a strut on the overall behaviour of the building. In each pushover analysis of a computational model, initially the gravity loads were applied in a force controlled manner. Next, the lateral loads were applied in a dis- (b) Using proposed hinge property placement controlled manner. The distribution of the lat- Fig. 7 Formation of hinges in a typical frame of Building I eral force along the height of the building was calculated as per a parabolic relationship [7]. A brief description of the buildings and the corresponding results are presented. Case Studies Building I Two multi-storeyed RC framed buildings were analysed to study the effect of the proposed hinge property for an Figure 6 shows the typical ﬂoor plan and frame elevation equivalent strut. One is a model regular building that was of the four storeyed building, using line diagram for each designed for seismic loads (Building I). The other one is an frame member. The concrete slab is 120 mm thick at every existing irregular building that was not designed to resist ﬂoor. Beams in all the ﬂoors are identical (230 mm earthquake forces (Building II). For each building, a width 9 500 mm depth). The size of the corner columns is computational model was developed using frame elements 400 9 350 mm . All the other interior columns are for the beams and columns, and strut elements for the inﬁll 350 9 350 mm . A raft foundation was assumed at the walls. The diaphragm action of the slabs was modelled by base of the columns. All the frames are ﬁlled with masonry assigning horizontal rigid diaphragm constraints. For a walls. The wall thicknesses are 230 and 120 mm for the frame element, a tri-linear ﬂexural hinge property near the exterior and interior walls, respectively. The compressive joint was idealised from the moment versus curvature strength of concrete and the yield stress for the reinforcing curve for the corresponding section of the member, and an steel were considered to be 20 and 415 MPa, respectively. estimate of the length of plastic hinge for calculating The compressive strength of the masonry units (burnt clay rotation. The moment versus curvature curve was devel- bricks) was taken as 3 MPa. oped based on nonlinear mechanics of RC, satisfying the In the pushover analysis using linear hinge properties for principles of equilibrium of forces at the section, compat- the struts, it was found that the formation of the hinges in ibility of strains of concrete and the reinforcing bars, and the struts leads to the termination of the pushover analysis. Base shear, kN 106 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 Fig. 9 Typical plan and elevation of Building II 3m 25.86 3m Stair case Lift core 3m 3m 3m 19.56 m (b) Sectional elevation along A-A (a) Floor plan Figure 7a shows the formation of hinges in a typical inte- of each level, to accomodate the lift core and the staircase, rior frame, using linear hinge properties for the struts. No two different diaphragm constraints were assigned at each hinge has formed in the beams or columns. This result is level. The dimensions of the beams and columns vary as not desirable, because in a pushover analysis the objective per the location. The columns are supported on isolated is to study the behaviour of the building with the sequential footings. Some of the frame panels are inﬁlled with con- formation of hinges in the frame members, even beyond crete hollow block masonry units. Only the walls without the failure of the struts. substantial openings were considered in the computational Figure 7b shows the formations of hinges in the same model. The compressive strength of concrete and the yield frame using the proposed hinge properties. The formation stress for the reinforcing steel were considered to be 25 and of hinges in the struts was followed by the formation of 415 MPa, respectively. The compressive strength of the hinges in the frame members. This is realistic and meets masonry units was taken as 1.65 MPa. the objective of the pushover analysis. Figure 8 compares Similar to that in Building I, a pushover analysis using the pushover curves using the two types of hinge properties linear hinge properties for the struts, got abruptly termi- for the struts. The pushover curve based on linear hinge nated. The soft-storey behaviour was not reﬂected in the property shows lower inelastic drift compared to that based analysis (Fig. 10a). However, when the analysis was on the proposed hinge property. The curve based on the repeated using the proposed hinge properties, even after the proposed hinge property reﬂects the effect of the modelled formation of hinges in the struts, there was formation of nonlinearity in the equivalent struts, and provides better hinges in the columns which revealed the soft-storey estimate of the deformation demand in each frame member behaviour of the building (Fig. 10b). Figure 11 compares as per the state of hinge formed in it. Also, the inelastic the pushover curves using the two types of hinge properties drift of the building leads to an appreciable lateral drift for the struts. (P - D) effect. The analysis results can be further studied in terms of the performances of the buildings and the frame members. Building II Davis et al. [4] used the proposed hinge property for the struts to analyse the performances of several open ground Figure 9 shows a typical ﬂoor plan and sectional elevation storey buildings. of an existing ﬁve-storeyed residential building. In the absence of inﬁll walls to accommodate parking of vehicles, the ground storey is a soft and weak storey, generating a Conclusion vertical irregularity in the building. As per the construction drawings, the concrete slab is 120 mm thick at every ﬂoor. In a pushover analysis of a framed building, for modelling Since there were large cut-outs near the centre of the slab the inﬁll walls by equivalent struts, it is necessary to assign 123 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 107 an appropriate axial load versus deformation hinge prop- erty to each strut. If a linear elastic relationship is used, then the formation of hinges in the struts may lead to ter- mination of the pushover analysis. This will not reﬂect an appropriate nonlinear behaviour of the building. On the contrary, when a nonlinear (simpliﬁed as piece-wise linear) relationship is used, the formation of hinges in the struts is followed by the formation of hinges in the frame members. This is realistic and meets the objective of the pushover analysis. Also, the pushover curve shows higher inelastic drift due to the modelled nonlinearity in the equivalent strut. Hence, it is recommended to use the proposed axial load versus deformation hinge property for the equivalent struts to model the effect of inﬁll walls without openings. The proposed hinge property can also be used as the backbone curve for modelling the hysteretic behaviour of (a) Using linear hinge property an equivalent strut in a nonlinear time history analysis of a IO CP LS building. Further reﬁnement is possible for the descending branch of the hinge property, based on deformation con- trolled tests. References 1. Achintya, P. Dayaratnam, S.K. Jain, Behaviour of brick inﬁlled RC frame under lateral load. Indian Concr. J. 65(9), 453–457 (1991) 2. H. Achyutha, S.S. Injaganeri, S. Satyanarayanan, C.S. Krishna- moorthy, Inelastic behaviour of brick inﬁlled reinforced concrete frames. J. Struct. Eng. 21(2), 107–115 (1994) 3. P.G. Asteris, S.T. Antoniou, D.S. Sophianopoulos, C.Z. Chryso- stomou, Mathematical macromodeling of inﬁlled frames: state of the art. J. Struct. Eng. 137(12), 1508–1517 (2011) 4. Davis, R., Menon, D. and Prasad, A. M., 2008, Evaluation of magniﬁcation factors for open ground storey buildings using nonlinear analyses, Proceedings, 14th World Conference on Earthquake Engineering, Beijing, P. R. China, October 12–18 5. FEMA 356, 2000, Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Federal Emergency Management (b) Using proposed hinge property Agency, USA 6. M. Holmes, Steel frames with brick and concrete inﬁlling. Proc. Fig. 10 Formation of hinges in the frame along A–A of Building II Inst Civil Eng. 19(4), 473–478 (1961) 7. IS 1893: 2002, Indian standard criteria for earthquake resistant design of structures, Part—1. General provisions and buildings, Bureau of Indian Standards, New Delhi 8. T.C. Liauw, K.H. Kwan, Uniﬁed plastic analysis for inﬁlled frames. J. Struct Eng. 111(7), 427–1448 (1985) 9. A.B. Mehrabi, P.B. Shing, M.P. Schuller, J.L. Noland, Experi- mental evaluation of masonry-inﬁlled RC frames. J. Struct. Eng. 122(3), 228–237 (1996) 10. A.B. Mehrabi, P.B. Shing, Finite element modeling of masonry- Based on linear strut hinge property inﬁlled RC frames. J. Struct. Eng. 123(5), 604–613 (1997) Based on proposed strut hinge property 11. Polyakov, S. V., 1963, Masonry in framed buildings, Gods- udarstvenoe Isdatel’ stvo Literatury Po Stroidal stvui Architec- ture, Moscow, Translated by G. L. Cairns, National Lending Library for Science and Technology, U. K 0.00 0.02 0.04 0.06 0.08 12. A. Saneinejad, B. Hobbs, Inelastic design of inﬁlled frames. Roof displacement m J. Struct. Eng. 121(4), 634–650 (1995) 13. SAP, NL (Computers and structures incorporated, Berkeley, Fig. 11 Pushover curve along X-direction (Building II) 2000) Base shear, kN 108 J. Inst. Eng. India Ser. A (May–July 2013) 94(2):99–108 14. A.K. Sengupta, A. Asokan, Modelling of inﬁll walls for nonlinear 15. B.S. Smith, C. Carter, A method of analysis for inﬁlled frames. static analysis of buildings subjected to seismic loads. Indian Proc. Inst. Civil Eng. 44(1), 31–48 (1969) Concr. J. 82(2), 19–28 (2008) "
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